Transfer learning for concept drifting data streams in heterogeneous environments

Abstract

Learning in non-stationary environments remains challenging due to dynamic and unknown probability distribution. This issue is even more problematic when there is a lack of supervision data for a specific domain, making the use of labeled data from a related but different domain highly valuable. This paper addresses the streaming data classification and introduces a heterogeneous unsupervised domain adaptation method. To cover the uncertainty caused by the distribution discrepancy and concept drifting data, the proposed method prioritizes target domain data with the highest uncertainty, as they indicate changes in data distribution. It utilizes a fuzzy-based feature-level adaptation and optimizes parameters through accelerated optimization. Additionally, it employs instance selection in the source domain to identify qualified samples, further enhancing classification and adaptation. Three different settings of the proposed method have been configured, and five state-of-the-art methods have been selected as competing methods. Regarding different types of concept drift, various experiments taken from four benchmark datasets demonstrate the superiority of the proposed method in terms of accuracy and computational time. The Wilcoxon statistical test has been conducted to prove a meaningful distinction between the evaluation metrics results of the proposed method and the competing ones.

Appendices

Appendix I

This section introduces some definitions of the fuzzy theory briefly.

Definition 1

A fuzzy set $\overline{A }$ in a universe of discourse $X=\left\{x\right\}$ is defined as $\overline{A }=\left\{x\text{,}\mu \left(x|\overline{A }\right)\right\}$ where $\mu \left(x|\overline{A }\right)\subseteq [0\text{,}1]$ is the membership function of an element $x\text{,} \;x\in {\mathbb{R}}^{n}$ in a fuzzy set $\overline{A }$.

Definition 2

A fuzzy set $\overline{A }\left({a}_{1}\text{,}{a}_{2}\text{,}\dots \text{,}{a}_{n}\right)$ of ${\mathbb{R}}^{n}$ is called a fuzzy vector at non-fuzzy set $A\left({a}_{1}\text{,}{a}_{2}\text{,}\dots \text{,}{a}_{n}\right){\in {\mathbb{R}}}^{n}$ if its membership function $\mu$ fulfills the following properties.

• 1. $\mu \left({x}_{1}\text{,}{x}_{2}\text{,}\dots \text{,}{x}_{n}|\overline{A }({a}_{1}\text{,}{a}_{2}\text{,}\dots \text{,}{a}_{n})\right)=1$ is upper semi-continuous in ${\varvec{x}}=({x}_{1}\text{,}{x}_{2}\text{,}\dots \text{,}{x}_{n}){\in {\mathbb{R}}}^{n}$.

• 2. $\mu \left(({x}_{1}\text{,}{x}_{2}\text{,}\dots \text{,}{x}_{n})|\overline{A }({a}_{1}\text{,}{a}_{2}\text{,}\dots \text{,}{a}_{n})\right)=1$ if and only if $\left({x}_{1}\text{,}{x}_{2}\text{,}\dots \text{,}{x}_{n}\right)=({a}_{1}\text{,}{a}_{2}\text{,}\dots \text{,}{a}_{n})$.

• 3. ${\overline{A} }_{\alpha }=\left\{{\varvec{x}}|\mu \left({\varvec{x}}|\overline{A }\left({a}_{1}\text{,}{a}_{2}\text{,}\dots \text{,}{a}_{n}\right)\right)=\alpha \text{,} {\varvec{x}}{\in {\mathbb{R}}}^{n}\right\}$ is a compact convex subset of ${\mathbb{R}}^{n}$ for all $\alpha$ in [0,1].

Definition 3

The $\alpha$-cut of a fuzzy set $\overline{A }$ in $X$, is defined as for each $\alpha \in (0\text{,}1]$.

Definition 4

Given $N$ fuzzy sets, ${\overline{A} }_{1}\text{,}{\overline{A} }_{2}\text{,}\dots \text{,}{\overline{A} }_{N}$, an operator $R:\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)\to [0\text{,}1]$ indicates a fuzzy relation if the following properties are fulfilled.

• 1. Reflexivity property: $R\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=1\text{,} \forall {\overline{A} }_{i}$.

• 2. Symmetry property: $R\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=R\left({\overline{A} }_{j}\text{,}{\overline{A} }_{i}\right)\text{,} \forall {\overline{A} }_{i}\text{,}{\overline{A} }_{j}$

Clearly, the fuzzy relation $R$ is an $N\times N$ matrix ${R}^{M}=\left({r}_{ij}\right)\text{,} {r}_{ij}=R\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)$.

Definition 5

The max–min composition of two fuzzy relations ${R}_{a}^{M}$ and ${R}_{b}^{M}$ is defined as:

$${\left({R}_{a}^{M}\circ {R}_{b}^{M}\right)}_{ij}=\underset{k=1}{\overset{N}{\bigvee }}\left({r}_{ik}^{(a)}\bigwedge {r}_{kj}^{(b)}\right)$$

(I-1)

where ${r}_{ik}^{(a)}$ is the element of ${R}_{a}^{M}$, ${r}_{kj}^{(b)}$ is the element of ${R}_{b}^{M}$, $\bigvee$ represents the maximum operation, and $\wedge$ denotes the minimum operation.

Definition 6

Given $N$ fuzzy sets, ${\overline{A} }_{1}\text{,}{\overline{A} }_{2}\text{,}\dots \text{,}{\overline{A} }_{N}$, an operator $R:\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)\to [0\text{,}1]$ indicates a fuzzy equivalence relation if the following properties are fulfilled.

• 1. Reflexivity property: $R\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=1\text{,} \forall {\overline{A} }_{i}$.

• 2. Symmetry property: $R\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=R\left({\overline{A} }_{j}\text{,}{\overline{A} }_{i}\right)\text{,} \forall {\overline{A} }_{i}\text{,}{\overline{A} }_{j}$

• 3. Transitivity property: ${R}_{(2)}^{M}={R}_{(1)}^{M}\circ {R}_{(1)}^{M}$.

where ${R}^{M}$ is the fuzzy relation matrix $R$ and $\circ$ is the max–min operator. Using fuzzy relations, a fuzzy equivalence relation is constructed.

Definition 7

Given $N$ fuzzy sets, ${\overline{A} }_{1}\text{,}{\overline{A} }_{2}\text{,}\dots \text{,}{\overline{A} }_{N}$, there must be a finite $m\in {\mathbb{Z}}$, and an operator ${R}_{T}$ holds the following conditions:

$$ R_{T}^{M} = R_{\left( m \right)}^{M} = \underbrace {{R^{M} \circ R^{M} \circ \ldots \circ R^{M} }}_{m} $$

$${R}_{T}^{M}={R}_{T}^{M}\circ {R}_{T}^{M}$$

where ${R}^{M}$ is the fuzzy relation matrix $R$, $\circ$ is the max–min operator, and ${R}_{T}^{M}$ is the fuzzy relation matrix of ${R}_{T}$. ${R}_{T}$ is the max–min transitive closure of $R$.

Appendix II

This section contains some detailed information for computing the metric $\mathcal{D}$ introduced by [61].

For each fuzzy feature vector ${\overline{A} }_{i}\left({a}_{i1}\text{,}{a}_{i2}\text{,}\dots \text{,}{a}_{in}\right)\in F({\mathbb{R}}^{n})$. For each ${\overline{a} }_{ij}\in F({\mathbb{R}})$, its membership function is computed by Eq. (II-1).

$${\mu }_{ij}(x|{\overline{a} }_{ij})=\left\{\begin{array}{l}0\; \forall x<{a}_{ij}-{\rho }_{i} \\ 1-\frac{{\Vert x-{a}_{ij}\Vert }_{1}}{{\rho }_{i}} \forall {\Vert x-{a}_{ij}\Vert }_{1}\le {\rho }_{i}\text{,} \, x\in R \\ 0 \forall \;x>{a}_{ij}+{\rho }_{i}\end{array}\right.$$

(II-1)

where ${a}_{ij}$ indicates the $j$ th feature value of the $i$ th sample and ${\rho }_{i}$ shows the hesitation degree of the $i$ th sample considering a triangular membership function. Utilizing Eq. (II-2), ${\mu }_{ij}({\varvec{x}}|{\overline{A} }_{i})$ where ${\varvec{x}}=({x}_{1}\text{,}{ x}_{2}\text{,}\dots \text{, }{x}_{n})$ is obtained.

$${\mu }_{ij}({\varvec{x}}|{\overline{A} }_{i})= \left\{\begin{array}{l}0\; if \exists {x}_{j}\text{,} { x}_{j}<{a}_{ij}-{\rho }_{i} \\ 1-\frac{{\Vert {\varvec{x}}-{a}_{ij}\Vert }_{1}}{{n\rho }_{i}} \;if \forall {x}_{j}\text{,} {\Vert x-{a}_{ij}\Vert }_{1}\le {\rho }_{i}\text{,} x\in {\mathbb{R}}^{n} \\ 0\; if \exists {x}_{j}\text{,} { x}_{j}>{a}_{ij}+{\rho }_{i}\end{array}\right.$$

(II-2)

To define the fuzzy relation between two heterogeneous domains (source and target), the following metric is defined to measure the distance between the fuzzy vectors:

$$\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=\frac{1}{n}\underset{0}{\overset{1}{\int }}sup\left\{{\mathcal{D}}_{\lambda }\left(u\text{,}v\right): {\mathcal{D}}_{\lambda }\left(u\text{,}v\right)\in\Omega \left(\lambda \right)\right\}d\lambda ; u\in {\overline{A} }_{i}\left(\lambda \right)\text{,} v\in {\overline{A} }_{j}(\lambda )$$

(II-3)

where $\lambda$ is the membership value, ${\mathcal{D}}_{\lambda }\left(u\text{,}v\right)$ indicates the distance between points $u$ and $v$ in ${\mathbb{R}}^{n}$ with the given $\lambda$. $\Omega \left(\lambda \right)$ is computed by Eq. (II-4).

$$\Omega \left(\lambda \right)=\left\{d(u\text{,}{\overline{A} }_{j}(\lambda ))\right\}\cup \left\{d(v\text{,}{\overline{A} }_{i}(\lambda ))\right\}$$

$$d(u\text{,}{\overline{A} }_{j}(\lambda ))=min\left\{d\left(u\text{,}v\right)\text{,} v\in {\overline{A} }_{j}(\lambda )\right\}$$

$$d(v\text{,}{\overline{A} }_{i}(\lambda ))=min\left\{d\left(v\text{,}u\right)\text{,} u\in {\overline{A} }_{i}(\lambda )\right\}$$

(II-4)

where $d\left(v\text{,}u\right)$ is the ${l}_{1}$-norm between two n-dimensional vectors ($u$ and $v$). Note that the supremum operator (sup) in Eq. (II-3) indicates the longest distance between the fuzzy vector of one specific domain to the fuzzy set of another domain. Eq. (II-3) can be rewritten as Eq. (II-5).

$$\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=\frac{1}{n}\underset{0}{\overset{1}{\int }}d\left({A}_{i}\text{,}{A}_{j}\right)+\frac{1}{2}\left|d\left(u\text{,}{\overline{A} }_{j}\left(\lambda \right)\right)+d(v\text{,}{\overline{A} }_{i}(\lambda ))\right|d\lambda $$

(II-5)

The above equation is de-fuzzified regarding Eqs. (II-1) and (II-2) as follows:

$$\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=\frac{1}{n}d\left({A}_{i}\text{,}{A}_{j}\right)+\frac{1}{2}\underset{0}{\overset{1}{\int }}\left|\left(1-\lambda \right){\rho }_{i}-\left(1-\lambda \right){\rho }_{j}\right|d\lambda $$

$$=\frac{1}{n}d\left({A}_{i}\text{,}{A}_{j}\right)+\frac{1}{2}{\Vert {\rho }_{i}-{\rho }_{j}\Vert }_{1}\underset{0}{\overset{1}{\int }}\left(1-\lambda \right)d\lambda $$

$$=\frac{1}{n}d\left({A}_{i}\text{,}{A}_{j}\right)+\frac{1}{4}{\Vert {\rho }_{i}-{\rho }_{j}\Vert }_{1}$$

(II-6)

Eq. (II-6) cannot be used directly for computing the fuzzy relation because it does not satisfy two properties of the fuzzy relation, including (1) symmetry, a condition in which $\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=\mathcal{D}\left({\overline{A} }_{j}\text{,}{\overline{A} }_{i}\right)\text{, }\forall {\overline{A} }_{i}\text{, }{\overline{A} }_{j}$ and (2) reflexivity, a condition in which $\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=1\text{,} \forall {\overline{A} }_{i}$. Thus, the following function is employed.

$${R}_{\mathcal{D}}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)={e}^{-\frac{\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)}{2{\sigma }^{2}}}$$

(II-7)